Summary: A proper incidentor coloring is called a \((k,l)\)-coloring if the difference between the colors of the final and initial incidentors ranges between \(k\) and \(l\). In the list variant, the extra restriction is added: the color of each incidentor must belong to the set of admissible colors of the arc. In order to make this restriction reasonable we assume that the set of admissible colors for each arc is an integer interval. The minimum length of the interval that guarantees the existence of a list incidentor \((k,l)\)-coloring is called a list incidentor \((k,l)\)-chromatic number. Some bounds for the list incidentor \((k,l)\)-chromatic number are proved for multigraphs of degree 2 and 4.